Properties

Label 2-927-103.27-c0-0-0
Degree $2$
Conductor $927$
Sign $0.826 - 0.563i$
Analytic cond. $0.462633$
Root an. cond. $0.680171$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.982 − 0.183i)4-s + (−0.404 + 1.42i)7-s + (−0.538 + 1.89i)13-s + (0.932 − 0.361i)16-s + (0.658 − 0.600i)19-s + (−0.602 − 0.798i)25-s + (−0.136 + 1.47i)28-s + (−0.380 − 0.981i)31-s + (1.01 − 1.63i)37-s + (−0.840 − 1.35i)43-s + (−1.00 − 0.623i)49-s + (−0.181 + 1.95i)52-s + (1.12 + 1.48i)61-s + (0.850 − 0.526i)64-s + (−1.91 + 0.544i)67-s + ⋯
L(s)  = 1  + (0.982 − 0.183i)4-s + (−0.404 + 1.42i)7-s + (−0.538 + 1.89i)13-s + (0.932 − 0.361i)16-s + (0.658 − 0.600i)19-s + (−0.602 − 0.798i)25-s + (−0.136 + 1.47i)28-s + (−0.380 − 0.981i)31-s + (1.01 − 1.63i)37-s + (−0.840 − 1.35i)43-s + (−1.00 − 0.623i)49-s + (−0.181 + 1.95i)52-s + (1.12 + 1.48i)61-s + (0.850 − 0.526i)64-s + (−1.91 + 0.544i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.826 - 0.563i$
Analytic conductor: \(0.462633\)
Root analytic conductor: \(0.680171\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (748, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 927,\ (\ :0),\ 0.826 - 0.563i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.178230333\)
\(L(\frac12)\) \(\approx\) \(1.178230333\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 + (-0.850 - 0.526i)T \)
good2 \( 1 + (-0.982 + 0.183i)T^{2} \)
5 \( 1 + (0.602 + 0.798i)T^{2} \)
7 \( 1 + (0.404 - 1.42i)T + (-0.850 - 0.526i)T^{2} \)
11 \( 1 + (0.982 - 0.183i)T^{2} \)
13 \( 1 + (0.538 - 1.89i)T + (-0.850 - 0.526i)T^{2} \)
17 \( 1 + (-0.273 + 0.961i)T^{2} \)
19 \( 1 + (-0.658 + 0.600i)T + (0.0922 - 0.995i)T^{2} \)
23 \( 1 + (-0.982 - 0.183i)T^{2} \)
29 \( 1 + (-0.602 - 0.798i)T^{2} \)
31 \( 1 + (0.380 + 0.981i)T + (-0.739 + 0.673i)T^{2} \)
37 \( 1 + (-1.01 + 1.63i)T + (-0.445 - 0.895i)T^{2} \)
41 \( 1 + (-0.602 + 0.798i)T^{2} \)
43 \( 1 + (0.840 + 1.35i)T + (-0.445 + 0.895i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.0922 + 0.995i)T^{2} \)
59 \( 1 + (-0.850 + 0.526i)T^{2} \)
61 \( 1 + (-1.12 - 1.48i)T + (-0.273 + 0.961i)T^{2} \)
67 \( 1 + (1.91 - 0.544i)T + (0.850 - 0.526i)T^{2} \)
71 \( 1 + (0.602 - 0.798i)T^{2} \)
73 \( 1 + (-0.646 + 0.322i)T + (0.602 - 0.798i)T^{2} \)
79 \( 1 + (-0.243 + 0.489i)T + (-0.602 - 0.798i)T^{2} \)
83 \( 1 + (-0.850 - 0.526i)T^{2} \)
89 \( 1 + (-0.932 - 0.361i)T^{2} \)
97 \( 1 + (0.726 - 0.961i)T + (-0.273 - 0.961i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.31229078357398417528650601124, −9.386669851776245709172755531877, −8.949939035679379301823474717624, −7.67406832008584808452935277914, −6.86889184651880828296963177723, −6.10354223059924382252062242266, −5.34691529868382475109355293963, −4.04111736000831319067966682926, −2.60137446843220795654745994309, −2.03167008809096728319065461568, 1.28092566477451443098327616928, 2.99120654652935979644801935729, 3.55256096737886480277565002443, 4.98246359182416361549919548656, 6.00670114230848600890501084261, 6.91855474870432141497861705302, 7.67801156681774777784383009377, 8.104216446246418142493810947789, 9.783852701195162210897998869796, 10.17394763133797488506124863842

Graph of the $Z$-function along the critical line